Lawson, Andrew
(2017)
Exploratory lattice QCD studies of rare kaon decays.
*University of Southampton, Doctoral Thesis*, 192pp.

## Abstract

The rare kaon decays K → πl^{+}l^{-} and K → πνν proceed via flavour changing neutral currents, and are thus heavily suppressed in the Standard Model. This natural suppression makes these decays sensitive to the effects of potential new physics. These decays first arise as second-order electroweak processes, hence we are required to evaluate four-point correlation functions involving two effective operators. The evaluation of such four-point correlation functions presents two key difficulties: the appearance of unphysical terms in Euclidean-space correlators that grow exponentially as the operators are separated, and the presence of ultra-violet divergences as the operators approach each other. I present the results of the first exploratory studies of the calculation of the long-distance contributions to these decays using lattice QCD. The decays K → πl^{+}l^{-} are completely long-distance dominated; this lattice calculation is thus the first step in providing ab-initio estimates for the amplitudes of these decays. Our simulations are performed using the 24^{3 }x 64 domain wall fermion ensemble of the RBC-UKQCD collaboration, with a pion mass of 430(2)MeV, a kaon mass of 625(2)MeV, and a valence charm mass of 543(13)MeV. In particular we determine the form factor, V (z), of the K^{+} (k) → π^{+} (p) l^{+}l^{-} decay from the lattice at small values of z = q^{2}=M^{2}_{K} (where q = k - p), obtaining V (z) = 1.37(36); 0.68(39); 0.96(64) for the three values of z = -0.5594(12), -1.0530(34), -1.4653(82) respectively.The decays K^{+} → π^{+}νν are short-distance dominated, although the long-distance contributions represent significant sources of uncertainty. The lattice calculation of the decay amplitudes is made particularly difficult by the presence of ultra-violet divergences in the four-point correlators. I present the calculation of the renormalised decay amplitudes, using the 16^{3}x32 domain wall fermion ensemble of the RBC-UKQCD collaboration, with a pion mass of 421(1)(7)MeV, a kaon mass of 563(1)(9)MeV, and a valence charm mass of 863(24)MeV. In particular we find the difference between the perturbative and lattice estimates of the charm contribution to these decays to be ΔP_{c} = 0.0040(13)(32)(-45).

**Final thesis - Version of Record**

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